Question

Finding unique root of polynomial using remainder theorem

Answer 1

The unique root of the polynomial (P(x) = 2x³ - 5x² + 3x - 7) is (x = 1).

Step-by-step explanation:

To find the unique root using the remainder theorem, we substitute (x = 1 into the polynomial (P(x)). If the result is zero, then x = 1 is a root. In this case, (P(1) = 2(1)³ - 5(1)² + 3(1) - 7 = 0), confirming that (x = 1) is indeed a root.

The remainder theorem states that if P(a) = 0, then (x - a) is a factor of (P(x). In our case, since P(1) = 0), (x - 1) is a factor. We can then perform polynomial division to factorize P(x) and find the remaining quadratic factor. The result is
`\(P(x) = (x - 1)(2x^2 - 3x + 7)\).`

Now, looking at the quadratic factor (2x² - 3x + 7), we can apply the quadratic formula to find its roots. The discriminant (b^2 - 4ac) is negative in this case, indicating that the quadratic factor has complex roots. Therefore, the only real root of the original polynomial (P(x)) is (x = 1), making it the unique root.